Mathematics / advanced
Probability on measurable spaces
Connect events, random variables, expectation, independence, and convergence through a rigorous probability framework.
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Module 01 advanced
Probability on measurable spaces
Connect events, random variables, expectation, independence, and convergence through a rigorous probability framework.
- 01
Probability spaces and random elements
Build a probability model by specifying outcomes, measurable questions, and a normalized measure before introducing random observations.
30 min - 02
Distributions as pushforward measures
Transport probabilities through measurable maps, compute transformed laws, and distinguish marginal distributions from joint information.
30 min - 03
Independence and product probabilities
Recognize independence as factorization of measurable information, build product laws, and distinguish mutual independence from pairwise tests.
30 min - 04
Expectation, integrability, and legitimate limit exchanges
Define expectations through positive and negative parts, connect moment assumptions to tail behavior, and choose the right convergence theorem.
30 min - 05
Borel–Cantelli lemmas and tail events
Turn probability bounds into almost-sure eventual statements and distinguish summable errors from independent repeated opportunities.
30 min - 06
Modes of probabilistic convergence
Distinguish pathwise, probabilistic, moment, and distributional limits, and test the implications using rare events.
28 min - 07
Characteristic functions and convolution
Encode probability laws with bounded complex exponentials and turn independent sums into products.
28 min - 08
Laws of large numbers
Explain why averages stabilize, distinguish weak and strong laws, and identify when dependence or nonintegrability defeats averaging.
28 min - 09
Weak convergence and the Portmanteau theorem
Translate weak convergence into usable statements about events while respecting boundaries, atoms, and unbounded observables.
28 min - 10
Central limit theorems and tightness
Explain Gaussian fluctuations, the role of normalization and characteristic functions, and why tightness alone does not imply a central limit theorem.
34 min